# Asymptotic likelihood of chaos for smooth families of circle maps - Mathematics > Dynamical Systems

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Abstract: We consider a smooth two-parameter family $f {a,L}\colon\theta\mapsto\theta+a+L\Phi\theta$ of circle maps with a finite number of critical points.For sufficiently large $L$ we construct a set $A L^{\infty}$ of $a$-values ofpositive Lebesgue measure for which the corresponding $f {a,L}$ exhibits anexponential growth of derivatives along the orbits of the critical points. Ourconstruction considerably improves the previous one of Wang and Young for thesame class of families, in that the following asymptotic estimate holds: theLebesgue measure of $A L^{\infty}$ tends to full measure in $a$-space as $L$tends to infinity.

Author: Hiroki Takahasi

Source: https://arxiv.org/